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[MATH] in the [MATH] -parameter plane in Fig. 2. There are two special points labeled as [MATH] and [MATH] on [MATH] that correspond to the so-called inclination-flip (IF) bifurcation of codimension-two |
Shilnikov et al. 2001 . Its feature is that it gives rise to instant homoclinic chaos in the phase space and to complex bifurcation structures in the parameter space of the system. With our new computational-symbolic toolkit we can clearly and quickly identify such bifurcations and their fine organizations in the param... |
First, we define a formal power series [MATH] for a finite binary sequence [MATH] of length [MATH] , after omitting the first [MATH] symbols for initial transients of the separatrix [MATH] or any other trajectory, as follows: |
[EQUATION] By construction, the range of [MATH] is , including the sequences [MATH] and [MATH] , resp., in the limit as [MATH] . For example, [MATH] for the aperiodic sequence [MATH] |
[MATH] in Fig. 1b, with [MATH] and [MATH] , is given by: [MATH] The [MATH] -quantities are used as invariants to discriminate or conjugate finite progressions of the separatrix [MATH] of the saddle against each other to identify and trace down corresponding bifurcation curves in the parameter space. Moreover, the quant... |
[MATH] for [MATH] and [MATH] ) and the primary homoclinic ( [MATH] ) bifurcations all originating from the codimension-2 Bogdanov-Takens point ( [MATH] Shilnikov et al. 2001 . Fig. 1f shows how the primary homoclinic loop transmutes into a double one along the curve [MATH] . The sweep reveals the way the inclination-fl... |
[MATH] and [MATH] points give rise to jets of homoclinic bifurcation curves spiraling to various self-similar cod-2 Bykov terminal T-points, including [MATH] and [MATH] corresponds to heteroclinic connections linking the saddle [MATH] with saddle-foci [MATH] [MATH] (Fig. 1e) and generating periodic sequences [MATH] [MA... |
The ( [MATH] )-sweep of [2-9]-length in Fig. 4 demonstrates the intrinsic re-arrangement of the bifurcation constituents of complexity for a different cut at [MATH] . Here, the secondary inclination-flip point, ( [MATH] ), gives rise to loci of outgoing homoclinic curves that being re-directed by a saddle point ( [MATH... |
[MATH] crosses over the stability boundary near [MATH] , so that both ends of the semi-annular structures merge to complete spirals around [MATH] (as demonstrated in Suppl. Movie 2.) Meanwhile, T-points [MATH] and [MATH] merge with the saddle [MATH] to transform into concentric cycles. |
These structures in the 2D sweeps are the contour curves of the corresponding surfaces in the 3D [MATH] -parameter space of model (1). Figure 5 shows its near this saddle, which is the critical point of of the 2D surface shaped as a hyperbolic paraboloid. Depending on the particular [MATH] -cuts the contour lines of th... |
While a detailed sweep for short-term transient dynamics lets us reveal the underlying homoclinic bifurcations, longer sweeps, omitting initial transients, are designed to localize stability windows corresponding to regular dynamics [of Morse-Smale systems] and regions of chaotic dynamics in the parameter space. We imp... |
Figure 6 represents the bi-parameter long sweeps of [1000-1999]-length to identify regions where the dynamics of model (1) is simple and complex, where insets a/c and b/d represent the PC- and LZ-algorithm based sweeps, respectively. Regions of solid monotone colors correspond to the stability windows with stable equil... |
Other future enhancements for the DSP-toolkit are to include the search algorithms for bifurcations of equilibrium states and periodic orbits such as period-doubling. |
In conclusion, we have demonstrated the proficiency of the new symbolic toolkit for computational studies of both short-term transient and long-term solutions and to analyze the bifurcation mechanisms underlying the onset of chaotic and regular dynamics in the phase and parameter space of the given OPL model and simila... |
This work was in part funded by NSF grant IOS-1455527, RSF grant 14-41-00044 at the Lobachevsky University of Nizhny Novgorod, RFFI grant 11-01-00001, and MESRF project 14.740.11.0919. We thank GSU Brain and Behaviors Initiative for the fellowship and pilot grant support and are grateful to all members of Shilnikov’s N... |
# Source: arxiv 1806.01315 # Title: Cell Motility Dependence on Adhesive Wetting # Sections: all # Downloaded: 2026-03-02T08:44:13.365219+00:00 |
Cell Motility Dependence on Adhesive Wetting Abstract Adhesive cell-substrate interactions are crucial for cell motility and are responsible for the necessary traction that propels cells. These interactions can also change the shape of the cell, analogous to liquid droplet wetting on adhesive substrates. To address how... |
Introduction Migration of eukaryotic cells plays an important role in many biological processes including development Munjal and Lecuit ( 2014 , chemotaxis Kölsch et al. 2008 , and cancer invasion Wirtz et al. 2011 . Cell migration is a complex process, involving external cues, intra-cellular biochemical pathways, and ... |
Here we investigate the dependence of motility on cell-substrate adhesion using a mathematical model in which we can alter the adhesive forces independent of frictional forces. We carry out numerical simulations of this model using the phase field approach, ideally suited for objects with deforming free boundaries Kock... |
II Results II.1 Model Our vertical cross-sectional model cell captures the interaction of the cell with the bottom and, possibly, top substrate, as well as the interior of the cell Tjhung et al. 2015 (Fig. ). This is in contrast to most computational studies of cell motility which model a flat cell that is entirely in ... |
Barnhart et al. 2011 ); Bois et al. 2011a ); Shao et al. 2012 ); Camley et al. 2013 ); Goff et al. 2017 Friction is caused by the motion of the cytoskeleton relative to the substrate and is taken to be proportional to the actin fluid velocity. To accurately capture cell shape and its deformations, we use the phase fiel... |
Shao et al. 2010 2012 ); Camley et al. 2013 ); Najem and Grant ( 2013 ); Ziebert and Aranson ( 2013 ); Alonso et al. 2018 In our model, boundary motion is driven by fluid flow which is determined by adhesion, friction, membrane forces and active protrusion. The cell is placed on a substrate which is parallel to the [MA... |
[EQUATION] where the advection term couples the velocity field of the actin fluid, [MATH] , to the phase field, [MATH] is the width of the boundary, [MATH] is a relaxation coefficient, [MATH] is a double-well potential with minima at [MATH] and [MATH] , and [MATH] is the local curvature of the boundary (see Supplementa... |
The actin fluid velocity field is determined by the stationary Stokes equation with an assumption of perfect compressibility (zero pressure and neglecting the inertial term because of low Reynolds number) Rubinstein et al. 2009 ); Bois et al. 2011b |
[EQUATION] where [MATH] is the viscosity of the cell and where [MATH] is the active stress due to actin polymerization, further detailed below. |
[MATH] is the interaction between the cell and substrate and contains both adhesion and friction, [MATH] . The adhesive force is given by [MATH] , with the cell-substrate interaction potential: |
[EQUATION] Here, [MATH] is a constant field which marks the substrate (or ceiling) and continuously changes from [MATH] (within the substrate) to [MATH] (out of substrate; Fig. S1). [MATH] is a potential with a negative adhesion energy per unit length controlled by a parameter [MATH] such that larger values of [MATH] r... |
The second term in [MATH] describes the frictional force between the cell and the substrate. Depending on the cell type, these forces can arise from focal adhesions or from non-specific cell-substrate interactions. For simplicity, the frictional force in our cross-sectional model is modeled as a viscous drag proportion... |
[EQUATION] where the first term is the cell-substrate friction, parameterized by the coefficient [MATH] , and the second term represents a damping force, introduced to increase numerical stability. We have verified that the cell speed changes little when we vary the drag coefficient [MATH] (Fig. S2). Initially, we will... |
Polarization in our model is introduced through the polarization indicator [MATH] which is steering the actin polymerization. For simplicity, we have chosen [MATH] at the front half and [MATH] at the rear half of the cell. This corresponds to different actin promoter (e.g., Rac or Cdc42) distributions at the front and ... |
with width [MATH] and located a distance [MATH] away from the substrate (Fig. S1). By making the active stress proportional to [MATH] we restrict possible protrusions to a narrow band parallel to the substrate and in the cell front. This band is schematically shown in yellow in Fig. In addition, we localize the stress ... |
The expression for the stress is then given by: [EQUATION] Here, [MATH] is the protrusion coefficient, and [MATH] is the normal to the cell boundary. Note that our model does not include any possible feedback between substrate and stress generation. |
Our simulations are carried out as described previously Camley et al. 2013 and further detailed in the Supplemental Material where we also list the full set of equations. As initial conditions, our simulations start with polarized cells in which the distribution of [MATH] is asymmetric. The cell’s speed is tracked by [... |
and simulations are continued until a steady state has been achieved. Parameters values used in the simulations are given in Table S1. |
II.2 Simulation results and analysis We first investigate how cells move on a single substrate with different adhesion energies. For this, we solve the phase field equations for different values of the adhesion parameter [MATH] . Examples of resulting cell shapes are shown in Fig. while an example of the actin fluid ve... |
To provide insights into the relation between adhesion, cell shape and speed, we consider a simplified version of Eq. ( ), similar to the 1D model examined in Ref. Carlsson ( 2011 Since only asymmetric stress will contribute to the cell’s speed Tanimoto and Sano ( 2014 , we only need to take into account the viscosity,... |
[EQUATION] where [MATH] [MATH] is a friction coefficient taken to be spatially homogeneous, and [MATH] is the active stress which is 0 outside the cell. Boundary conditions include a steady cell shape [MATH] , zero net traction force [MATH] , and zero parallel stress [MATH] , where [MATH] are the normal and tangential ... |
It is in general not possible to solve Eq. in a arbitrary geometry. However, for the special case of a fixed-shape rectangular cell with length [MATH] and height [MATH] occupying [MATH] |
we can solve for the cell speed [MATH] (see the Supplemental Material). By averaging the stress over the vertical direction and following Carlsson’s one-dimensional solution Carlsson ( 2011 , we find: |
[EQUATION] where [MATH] determines the spatial scale of the decay of a point stress source Carlsson ( 2011 and where [MATH] (see also the Supplemental Material). From this solution it is clear that asymmetric active stress distribution will lead to cell motion. When [MATH] , corresponding to a highly viscous cytoskelet... |
[EQUATION] which shows that the cell speed scales inversely with the height of the cell, and that this scaling is independent of the cell length. Of course, a real cell will not be rectangular, and in the Supplemental Material we show that the cell speed scales with the average height for a more complex-shaped cell (Fi... |
Interestingly, the above found relation between cell speed and cell height does not depend on the way the cell’s effective height is altered. To verify this, we also simulated cells in confined geometries in which they are “squeezed” between two substrates, as shown in Fig. a (an example of a cell with the actin fluid ... |
Our results can be explained by realizing that cells contain a cytoskeleton network that can be described as a compressible viscous actin fluid. This actin fluid contains “active” regions which are confined to a layer with fixed width of [MATH] , and “passive” regions that are outside these active regions. Active stres... |
[MATH] and cell height constant. Consistent with our theoretical predictions, the speed of these cells is independent of the chamber height (Fig. S7). In addition, we have simulated cells in which the active stress region spans the entire front. Again in line with our theoretical insights, the cell speed was found to b... |
In our simulations, we have kept the friction coefficient constant and have thus ignored any potential link between adhesion and friction. This is likely appropriate for Dictyostelium cells but may not be valid for mammalian cells that have integrin mediated focal adhesions. The exact dependence of friction on adhesion... |
Different dependencies between friction and adhesion correspond to different trajectories through the two-dimensional phase space of Fig. a. The results we have presented so far correspond to traversing the phase space along the white dashed line in Fig. a. The black dashed line in this figure, on the other hand, repre... |
increases over orders of magnitude when adhesion [MATH] changes by small amounts. II.3 Experimental results To test the above predictions, we performed motility experiments of Dictyostelium discoideum cells. Importantly, these cells, unlike mammalian cells, do not make integrin mediated focal adhesions and their substr... |
Loomis et al. 2012 Experiments are carried out in microfluidic devices, as shown in Fig. a and modified from earlier work Skoge et al. 2014 (see also Supplemental Material and Fig. S9). Cells are moving in chambers with height [MATH] and with substrates that have variable adhesive properties. A constant cAMP gradient i... |
Dictyostelium cells move by extending actin filled protrusions called pseudopods which can extend over a significant distance from the substrate. As a consequence, our confined cells occlude the entire space between two substrates. This was verified explicitly by labeling the cell with a fluorescent membrane marker and... |
The top substrate of the chamber consists of Polydimethylsiloxane (PDMS) and the bottom substrate is either made of PDMS or is coated with a thin layer of Polyethylene glycol (PEG) gel. Cells moving on these PEG-coated substrates have vastly reduced adhesion, as reported in earlier studies Tzvetkova-Chevolleau et al. 2... |
Our theoretical predictions for cells in confinement are that decreased height increases speed, and that cells in asymmetric adhesion are faster than cells in symmetric adhesion. Both of these qualitative predictions are observed in our experiments. First, our experiments show that cell speed is significantly affected ... |
[MATH] m which, in turn, have smaller speed than cells in chambers with [MATH] m. The trend of slower motion in deeper chambers holds for both PDMS and PEG coated bottom substrates. Furthermore, we have verified that these results do not depend on the steepness of the gradient (Fig. S11). These observations are fully c... |
In addition, our experiments show that cells moving in a chamber with unequal top and bottom adhesion are markedly asymmetric (Fig. c), consistent with past results that showed that |
Dictyostelium cells only weakly adhere to PEG. Specifically, the contact area of cells on PEG coated substrates is significantly smaller than the contact area on PDMS substrates and the resulting asymmetry can be quantified by the ratio of bottom and top contact area. Cells with PDMS on top and bottom and for [MATH] [M... |
[MATH] [MATH] m have ratios close to 1 indicating that the shape is symmetric. In contrast, cells moving in chambers with these values of [MATH] that have a PEG bottom have ratios that are much smaller than 1, indicating a more asymmetric cell shape. For the largest value of [MATH] [MATH] 10 [MATH] m) cell preferential... |
Importantly, quantifying the cell speed for the different chambers reveals that cells in the symmetric PDMS/PDMS condition move slower than cells in the asymmetric PDMS/PEG condition (Fig. b). Again, these experimental results are fully consistent with our theoretical and numerical predictions and show that cell shape,... |
III Discussion and conclusion In this study, we examined how cell shape can affect cell speed using simulations, analytics, and experiments. We should stress that our experiments can only be compared to the simulations on a qualitative level. Values for the model parameters are not precisely known, and our model cell i... |
DiMilla et al. 1991 ); Palecek et al. 1997 ); Liu et al. 2015 Our results suggest that the increase of cell speed with increased adhesion found in these experiments might be attributed to cell spreading and a lower effective height. The observed decrease in cell speed following a further increase in adhesion can then b... |
Our numerical and experimental results indicate that changing cell morphology through confinement can also significantly alter the migration speed, with decreasing chamber heights resulting in increased cell speeds. Comparison with other cell types is challenging as cells might change their behavior following confineme... |
We should point out that the simple scaling of cell speed dependence on cell height (Eq. ) is based on the assumption of localized active stress (the numerator) and uniform cytoskeleton viscosity (the denominator) in the entire cross section. As shown in our experimental work and in previous studies Nagel et al. 2014 F... |
In summary, we show how adhesion forces result in cell spreading and that the accompanying shape changes can result in larger velocities. Key in this result is the existence of a narrow band of active stress that has a smaller spatial extent than the height of the cell. As a result, the dissipation due to the shear str... |
Supplemental Material for “Cell Motility Dependence on Adhesive Wetting” IV Phase field model of cell motility The equations for the phase-field cross section model are: |
[EQUATION] Here, [MATH] describes the field of the cell. The double-well potential is defined as [MATH] and the curvature is computed as [MATH] while |
[MATH] is a relaxation coefficient. The force terms are explicitly explained below. The substrate force contains the cell-substrate adhesion and friction: [MATH] , where |
[EQUATION] Here, [MATH] is the velocity field of the actin fluid and [MATH] are the cell-substrate friction coefficient and damping coefficient, respectively. [MATH] is the field describing the substrate, and [MATH] is the interaction potential between the cell and substrate. The The cell moves either on top of a plain... |
with a boundary width of [MATH] (Fig. S1 ). Here, [MATH] indicates the substrate into which the cell cannot penetrate, and [MATH] indicates the region accessible to the cell. In our simulations, the substrate is parallel to the x direction and, for the case of a single substrate located at [MATH] |
[MATH] is written as [EQUATION] For a chamber with a parallel top substrate located at [MATH] this becomes [EQUATION] Given [MATH] and [MATH] , the interaction potential is: |
[EQUATION] where [MATH] contains an attractive term, corresponding to adhesion, and a repulsive term, corresponding to the non-penetrability of the substrate. For the bottom substrate, we use |
[EQUATION] while the potential for the top substrate has an identical form with [MATH] replaced by [MATH] . Here, [MATH] is the adhesion energy per unit length, [MATH] is a parameter that measures the penalty of overlap between cell and substrate Camley et al. 2014 and [MATH] is a double-well potential [MATH] The energ... |
[EQUATION] corresponds, in the sharp interface limit, to an adhesive energy equal to [MATH] where [MATH] is the length of the cell in contact with the substrate. Note that the inclusion of the [MATH] results in a force that only vanishes outside the membrane Zhao et al. 2017 In our simulations we take [MATH] . For this... |
[MATH] we simulated cells without any propulsive force. The resulting static shapes can be directly compared to standard energy minimization simulations. Fig. S3 |
shows that the phase field shapes agree well with shapes obtained using Surface Evolver, a simulation tool that evolves surfaces toward minimal energy by a gradient descent method Brakke ( 1992 |
The contribution from both the tension and bending of the membrane is captured by [MATH] In our simulation we ignore the bending term since it contributes little to the shape of cell. The tension energy is given by Shao et al. 2010 ); Camley et al. 2013 |
[EQUATION] resulting in [MATH] Area conservation is introduced via [MATH] with [MATH] the prescribed area size, and [MATH] a parameter which controls the strength of the area constraint Shao et al. 2010 |
The active stress term in our model, [MATH] , is similar to our earlier work Shao et al. 2012 but only acts near the substrate. This is accomplished through the addition of the term |
[MATH] , where [MATH] , for the bottom substrate, takes on the form [EQUATION] A similar expression is used for the top substrate. The inclusion of [MATH] results in active stresses confined to a band with width [MATH] and located a distance [MATH] away from the substrate (Fig. S1 ). Note that vertical height of the ac... |
[MATH] and that [MATH] Three examples of the velocity fields obtained numerically are shown in Fig. S4 , corresponding to the cell motion on single substrate, confined in channels and confined in channels with asymmetric adhesion (Fig. 2 and Fig. 3 in main text). The retrograde flow patterns are similar to previous stu... |
Numerical Methods The equation for [MATH] is stepped by uniform time step [MATH] in a forward Euler scheme so that [MATH] at time step [MATH] is obtained from |
[MATH] at time step [MATH] [EQUATION] Here, [MATH] is computed using a finite difference method and all other differentiation operators are computed using a fast Fourier spectral method. Simulations were carried out on a [MATH] grid of size [MATH] Model parameters, modified from Shao et al. 2012 ); Camley et al. 2013 ,... |
The velocity field [MATH] is updated every time step by a semi-implicit Fourier spectral method after updating [MATH] as detailed in |
Camley et al. 2013 . The equation is iterated as: [EQUATION] where [MATH] , and [MATH] represents the terms in the Stokes equation that are independent of the iteration step [MATH] . The iteration will continue until |
[EQUATION] or until a maximal number of iterations (here chosen to be 20) is reached. VI Analytical Results As stated in the main text, we aim to analytically solve Eq. S1&S2, where several simplifications have to be made. First, we are trying to find the steady-state solutions, so the cell shape will not change with t... |
[EQUATION] where [MATH] is the cell’s mass of center velocity, which is our target to solve, and [MATH] is the normal unit vector of the boundary. The cell’s boundary is free so the parallel stress at the boundary is zero |
[EQUATION] where [MATH] is the tangential unit vector of the boundary. Notice that the active stress is always constrained inside the cell so it will not enter any boundary conditions. The total force of the cell exerted on substrate should be balanced which gives a zero net traction force condition |
[EQUATION] where [MATH] is the friction coefficient at different locations. To get analytical expressions, we neglect the spatial heterogeneity in friction and simply take [MATH] . This simplification does not change the central feature of our main result (the cell’s speed is inversely related to the cell’s height). |
Second, we only take into account the viscosity, friction and active stress because they are directly related to the cell motion. The adhesion, area conservation and membrane forces only contribute to the cell’s shape, which is implicitly included in the boundary conditions. Thus we get a simplified equation for Eq. S2... |
[EQUATION] Integrating the above equation and using the zero traction force condition, we obtain [MATH] . As the active stress [MATH] is constrained inside the cell, this will lead to a condition equivalent to the zero traction force condition |
[EQUATION] which is the zero traction force condition we used below. Notice that a fixed cell shape has to be given in order to apply the boundary conditions. Since we only care about the cell’s mass of center velocity [MATH] , and not the full solution for [MATH] , we will next show how [MATH] can be obtained without ... |
VI.1 Analytical solution of the rectangular model cell Here we wish to solve the Eq. S4 for a rectangular fixed cell shape [MATH] with an unknown cell speed [MATH] (notice we put the x-direction as cell moving direction so [MATH] is a scalar). The boundary conditions are [MATH] . Integrating the Stokes equation, we get... |
The tangential vector [MATH] can be determined by the normal vector [MATH] . The zero-parallel stress condition [MATH] results in |
[EQUATION] For rectangular boundaries, these conditions lead to [EQUATION] at all boundaries. Since the cell is moving along x-direction, only [MATH] is relevant and we can integrate the 2D Stokes equation in the y-direction. With the condition of [MATH] , we obtain a 1D Stokes equation: |
[EQUATION] where [MATH] , and [MATH] . The corresponding boundary conditions are [MATH] and [MATH] . This is exactly the same problem as in reference Carlsson ( 2011 . Using standard Green’s function methods, we obtain: |
[EQUATION] and, since [MATH] at boundaries [MATH] , we obtain [EQUATION] as reported in the main text (Eq. 5). If the active stress is confined in a band with width [MATH] , i.e., [MATH] , the cell’s speed [MATH] will scale as: |
[EQUATION] where [MATH] is a constant, corresponding to the boundary velocity determined by the 1D problem [MATH] with homogeneous boundary conditions. Notice that this scaling does not depend on the vertical position of the active stress. Therefore, our model will give the same cell speed independent of the type of ac... |
VI.2 Effective height for non-rectangular cells In the above section, the speed of a rectangular cell was determined exactly. Actual cells are, of course, not rectangular but obtaining a solution for cells with more complex shapes is challenging. Nevertheless, insight can be obtained by considering a cell composed of t... |
[EQUATION] To simplify the problem, we introduce the new variables [MATH] and [MATH] . Using the continuity condition we have: [EQUATION] |
Together with [MATH] we get [EQUATION] The zero traction force will give [EQUATION] Combining with the stress continuity we obtain |
[EQUATION] such that [EQUATION] Notice that Eq. S10 and Eq. S11 have clear physical meanings, namely flow conservation and force balance, respectively. It is convenient to introduce the net flow [MATH] and net force [MATH] on each rectangle: |
[EQUATION] and, using the zero-parallel stress condition, we obtain the 1D version of the problem for the right and left rectangle: |
[EQUATION] with [MATH] [MATH] can be solved by superposition of two parts: [MATH] with homogeneous boundary conditions and active stress, and [MATH] with inhomogeneous boundary conditions but zero active stress. After substituting [MATH] , we obtain |
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